CPSC 499 — Computational Ring Theory

Definition of rings. Examples. The integers. The reals. Fraction fields. Polynomial Rings. Exact division rings. Matrix rings. Formal power series. Modular arithmetic. Prime fields. Quotient rings.

Classification of Rings. Commutative rings. Integral domains. Euclidean domains. Principal Ideal Domains. Unique factorisation domains. Fields. Division rings.

The integers. Addition and subtraction for the integers. The basic multiplication and division algorithms for the integers. Kurasawa's algorithm. Using Newton's method for division.

Euclidean domains and Euclid's algorithm. Arithmetic for the rationals. The Chinese remainder theorem for integers.

Homomorphisms and ideals. The first isomorphism theorem for rings.

Polynomial rings. Lagrange interpolation. Evaluation homomorphisms. Determinants of matrices of univariant polynomials. Fast Fourier transforms and multiplication. Fast Fourier transforms and prime fields.

p-adic number fields. Hensel lifting.

The running time of Berlekamp's algorithm. Factorisation over Z[X].